Deck 12: Boolean Algebra

Question 1

Use a K-map to minimize the sum-of-products expansion x y z + x y z + x y z + x y z.

Answer 1

We construct the following K-map. Combining terms gives us the expansion

Question 2

What is the output of the following circuit?

Answer 2

The output of the circuit is

Question 3

Prove or disprove that x y + y = y whenever x and y are Boolean variables.

Answer 3

When y = 1 we have x y + y = x + 1 = 1 = y. When y = 0 we have x y + y = x · 0 + 0 = 0 + 0 = 0 = y. Hence x y + y = y for all values of the Boolean variables x and y.

Question 4

Use a K-map to minimize the sum-of-products expansion

Question 5

Find the sum-of-products expansion of a Boolean function f(x, y, z) that is 1 if and only if x = y = 1 and z = 0, or x = 0 and y = z = 1, or x = y = 0 and z = 1.

Question 6

Find a Boolean function f(x, y, z) that has the value 1 if and only if exactly two of x, y, and z have the value 1.

Question 7

What is the value of the Boolean function

Question 8

How many different Boolean functions are there of degree 3?

Question 9

Is the set of operators {+, ·} functionally complete? Justify your answer.

Question 10

Prove or disprove that whenever x, y, and z are Boolean variables.

Question 11

Construct a circuit using inverters, OR gates, and AND gates that gives an output of 1 if three people on a committee do not all vote the same.